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Monday 23 May 2016 by Maddle Stone @ 12:00 PM

a magic square is an arrangement of distinct numbers (i.e., each number is used once), usually integers, in a square grid, where the numbers in each row, and in each column, and the numbers in the main and secondary diagonals, all add up to the same number, called the “magic constant.” A magic square has the same number of rows as it has columns, and in conventional math notation, "n" stands for the number of rows (and columns) it has. Thus, a magic square always contains n2 numbers, and its size (the number of rows [and columns] it has) is described as being "of order n." A magic square that contains the integers from 1 to n2 is called a normal magic square. (The term "magic square" is also sometimes used to refer to any of various types of word squares.) Normal magic squares of all sizes except 2 × 2 (that is, where n
= 2) can be constructed. The 1 × 1 magic square, with only one cell
containing the number 1, is trivial. The smallest (and unique up to rotation and reflection) non-trivial case, 3 × 3, is shown below.

Any magic square can be rotated and reflected to produce 8 trivially
distinct squares. In magic square theory, all of these are generally
deemed equivalent and the eight such squares are said to make up a
single equivalence class.The constant that is the sum of every row, column and diagonal is called the magic constant or magic sum, M. Every normal magic square has a constant dependent on n, calculated by the formula M = [n(n2 + 1)] / 2. For normal magic squares of order n = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS). Magic squares have a long history, dating back to 650 BC in China. At
various times they have acquired magical or mythical significance, and
have appeared as symbols in works of art. In modern times they have been
generalized a number of ways, including using extra or different
constraints, multiplying instead of adding cells, using alternate shapes
or more than two dimensions, and replacing numbers with shapes and
addition with geometric operations.